Mathematics I Model question question paper

Solve the inequality $3+2x-x^2\ge 0$.

If a, b, c, and d are in G.P. prove that $a^2-b^2,b^2-c^2,c^2-d^2$ are also in G.P.

Prove that $$\left[\begin{array}{ccc}1+x& 1& 1\\ 1& 1+y& 1\\ 1& 1& 1+z\end{array}\right]=xy\left(\frac{1}{x}+\frac{1}{y}+\frac{1}{z}+1\right)$$

Find the equation of the ellipse whose latus rectum is 5 and the eccentricity is $\frac{1}{\sqrt{2}}$.

If $\vec{a},\vec{b}=\sqrt{3}$ and $\vec{a}\times \vec{b}=(1,2,2)$ find the angle between $\vec{a}$ and $\vec{b}$.

How many numbers of three different digits less than 500 can be formed from the integers 1, 2, 3, 4, 5, and 6?

Find the domain and range of the function $f(x)=\sqrt{9-x^2}$.

Write contrapositive of the statement "If you are I.T. student then you read mathematics".

Find absolute value of the complex number $\frac{3-4i}{3+4i}$.

What is the sum of the series $1+\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\frac{1}{16}$.

If A is a matrix of size $m\times n$ and B is a matrix of size $p\times q$ and multiplication BA is defined, then find the size of the matrix BA.

Write equivalent Cartesian equation of the polar equation $r\sin \theta =2$.

Write foci of the ellipse $\frac{x^2}{16}+\frac{y^2}{9}=1$.

For any two non-parallel vectors $\vec{a}$ and $\vec{b}$ write unit vectors perpendicular to both $\vec{a}$ and $\vec{b}$.

Write vector area of the parallelogram having two adjacent sides $\vec{a}$ and $\vec{b}$.

Write permutation of 12 numbers in a clock.

How many ways are there to arrange the word "PETROL", which do not begin with P?

Solve the inequality $3+2x-x^2\ge 0$.

If a, b, c, and d are in G.P. prove that $a^2-b^2,b^2-c^2,c^2-d^2$ are also in G.P.

Prove that $$\left[\begin{array}{ccc}1+x& 1& 1\\ 1& 1+y& 1\\ 1& 1& 1+z\end{array}\right]=xy\left(\frac{1}{x}+\frac{1}{y}+\frac{1}{z}+1\right)$$

Find the equation of the ellipse whose latus rectum is 5 and the eccentricity is $\frac{1}{\sqrt{2}}$.

If $\vec{a},\vec{b}=\sqrt{3}$ and $\vec{a}\times \vec{b}=(1,2,2)$ find the angle between $\vec{a}$ and $\vec{b}$.

How many numbers of three different digits less than 500 can be formed from the integers 1, 2, 3, 4, 5, and 6?

Find the domain and range of the function $f(x)=\sqrt{9-x^2}$.

a. Find Eigen values and corresponding Eigen vector of the matrix $A=\left[\begin{array}{cc}3& 1\\ 1& 3\end{array}\right]$ b. Let $$A=\left(\begin{array}{cc}1& -1\\ 3& 5\\ 2& 7\end{array}\right),u=\left(\begin{array}{c}1\\ 2\end{array}\right)$$ define a transformation $T:{\mathbb{R}}^2\to {\mathbb{R}}^3$, by $T(x)=Ax$ find $T(u)$.

a. Find sum of squares of the first n natural numbers. b. Find a vector and unit vector perpendicular to the plane of $P(1,-1,0)$, $Q(2,1,-1)$ and $R(-1,1,2)$.

a. Find the equation of a hyperbola in standard form having focus (-2, 0) and Diretrix $x=-\frac{1}{2}$. b. In an examination paper on mathematics, 20 questions are set. In how many different ways you can choose 18 questions to answer?